Q1: What is the correct Leibniz formula for the determinant of the matrix $$ \begin{pmatrix} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33} \ \end{pmatrix} $$ with real entries?

A1: $$ \sum_{(j_1, j_2, j_3) \in S_3} \mathrm{sgn}( (j_1, j_2, j_3) ) a_{j_1,1 } a_{j_2,2 } a_{j_3 ,3 } $$

A2: $$ \sum_{(j_1, j_2, j_3) \in S_3} \mathrm{sgn}( (j_1, j_2, j_3) ) a_{j_1,3 } a_{j_2,2 } a_{j_3 ,1 } $$

A3: $$ \sum_{(j_1, j_2, j_3) \in S_3} \mathrm{sgn}( (j_1, j_2, j_3) ) a_{j_1,3 } a_{j_3 ,1 } $$

A4: $$ \sum_{(j_1, j_2, j_3) \in S_3} (-1 )^{n} a_{j_1,1 } a_{j_3 ,2 } a_{j_2 ,3 } $$

Q2: What is the signum of the permutation $(1,2,3,4) \leadsto (1,4,3,2)$?

A1: $-1$

A2: $+1$

A3: $0$

A4: One needs more information.